Perfect subtree property for weakly compact cardinals

Yair Hayut; Sandra Müller

doi:10.1007/s11856-022-2385-4

Perfect subtree property for weakly compact cardinals

Yair Hayut, Sandra Müller^*

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We investigate the consistency strength of the statement: κ is weakly compact and there is no tree on κ with exactly κ⁺ many branches. We show that this statement fails strongly (in the sense that there is a sealed tree with exactly κ⁺ many branches) if there is no inner model with a Woodin cardinal. Moreover, we show that for a weakly compact cardinal κ the nonexistence of a tree on κ with exactly κ⁺ many branches and, in particular, the Perfect Subtree Property for κ, implies the consistency of AD_ℝ + DC.

Original language	American English
Pages (from-to)	865-886
Number of pages	22
Journal	Israel Journal of Mathematics
Volume	253
Issue number	2
DOIs	https://doi.org/10.1007/s11856-022-2385-4
State	Published - Mar 2023

Bibliographical note

Funding Information:
The second-listed author was supported by L’ORÉAL Austria, in collaboration with the Austrian UNESCO Commission and in cooperation with the Austrian Academy of Sciences — Fellowship Determinacy and Large Cardinals and Elise Richter grant number V844 of the FWF.

Funding Information:
The first-listed author was supported by Austrian Science Fund (FWF) Lise Meitner grant 2650-N35.

Publisher Copyright:
© 2022, The Hebrew University of Jerusalem.

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10.1007/s11856-022-2385-4

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title = "Perfect subtree property for weakly compact cardinals",

abstract = "We investigate the consistency strength of the statement: κ is weakly compact and there is no tree on κ with exactly κ+ many branches. We show that this statement fails strongly (in the sense that there is a sealed tree with exactly κ+ many branches) if there is no inner model with a Woodin cardinal. Moreover, we show that for a weakly compact cardinal κ the nonexistence of a tree on κ with exactly κ+ many branches and, in particular, the Perfect Subtree Property for κ, implies the consistency of ADℝ + DC.",

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doi = "10.1007/s11856-022-2385-4",

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AB - We investigate the consistency strength of the statement: κ is weakly compact and there is no tree on κ with exactly κ+ many branches. We show that this statement fails strongly (in the sense that there is a sealed tree with exactly κ+ many branches) if there is no inner model with a Woodin cardinal. Moreover, we show that for a weakly compact cardinal κ the nonexistence of a tree on κ with exactly κ+ many branches and, in particular, the Perfect Subtree Property for κ, implies the consistency of ADℝ + DC.

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Perfect subtree property for weakly compact cardinals

Abstract

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